Perhaps it says something about human psychology that if a theoretical model for some problem makes predictions — better yet, sensational or bizarre predictions — which later come out to be true, then we take the model more seriously. But should we? After all, other models may have made the same or similar predictions. Today, my intention isn’t to launch a debate on that contentious issue, but to retell two stories of romantic nostalgia that I have found fascinating.

It was 1869. An otherwise sane and supremely distinguished professor of chemistry in St. Petersburg had started a flood of chemical predictions, of discovering hitherto unknown substances on the face of the earth. So vivid were his predictions as to the appearance and properties of these ghostly substances, that apparently he was considered either a prophet or a madman. Numerical values for the atomic weights of the known elements had only been been known for about 30 years at this point. For more than 20 years, Dmitri Mendeléeff just played with this limited set of numbers, painstakingly arranging and rearranging the elements with their atomic weights. He found an incredible pattern in the weights of elements which were chemically equivalent or related. When he found gaps in the atomic weights of groups of equivalent elements, he predicted an undiscovered element that would fill that hole. Coincidence? Some obviously thought so: at a meeting of the English Chemical Society, the question was asked, sarcastically, whether he had considered arranging the elements according to the first letter of their names and looking for patterns. But six years later, de Boisbaudran discovered one of the predicted substances in the deep mines of Pyrenees, and every one of its properties matched Mendeléeff’s predictions: eka-aluminum was found on earth! One after another, came the vindications. Mendeléeff, we now know, was doggedly on the scent of the periodic law, today an established truth. Some of his predecessors had similar instincts, notably Lothar Meyer, Dumas and Strecker; but none had Mendeléeff’s determination and ability to ignore ridicule. A numerical law with very limited data boldly envisioned by an individual was eventually accepted as a remarkable scientific fact.

Mendeléeff carried with him other striking scientific ideas throughout his life, such as a substance which pervades the whole universe. He had, apparently, a uniquely ingenious theory on the origin of the fantastic oil deposits in Baku. He was summoned to Pennsylvania to help produce oil on a commercial scale for the first time. He was twice denied the Nobel prize, the story of which is now well known.

The other story I want to retell is that of Johann Bode, the German astronomer who noticed remarkable regularities in the planetary spacings in our solar system and put forth a mathematical formula for their locations. Bode’s formula says that the distances of the planets from the sun in their natural order follow the formula
α, α + β, α + 2β, α + 4β, α + 8β, … , with α = 0.4 and β = 0.3. The predicted positions of the planets by Bode’s formula are 0.4, 0.7, 1, 1.6, 2.8, 5.2, 10, 19.6, 38.8, 77.2, etc. The actual positions are 0.39 (Mercury), 0.72 (Venus), 1 (Earth), 1.52 (Mars), 2.77 (Asteroid belt), 5.20 (Jupiter), 9.54 (Saturn), 19.18 (Uranus), 30.06 (Neptune), 39.79 (Pluto). The good fit for Mercury, Venus, and Earth is not relevant; Bode found his formula by using the positions of those three planets. But it is impressive that the formula fits Jupiter exactly, even though Jupiter was an extrapolation. In 1772, when Bode gave his formula, the existence of the asteroids, Uranus, Neptune, and Pluto was not known. In 1781, musician-turned-astronomer William Herschel discovered a faint object moving very slowly around the sun and used Kepler’s third law to find its orbit. Uranus was discovered at a mean distance of 19.2, the exact hypothetical position predicted by Bode’s formula. King George III was ruling England then, and Herschel named the planet “Georginum Sidus.” Herschel was made a Fellow of the Royal Society, and given a King’s medal. (The name didn’t last long; it was renamed Uranus.) The coincidences continue. On the night of January 1, 1800, Giuseppe Piazzi found a new object between Mars and Jupiter, and then lost it. Heinrich Olbers happened to know of the newly found least squares method and used it to predict the position of the lost object. And, there it was, when the winter cleared: the asteroid belt, at almost exactly the Bode-predicted distance. But, of course, Bode didn’t predict any planet at Neptune’s current position, which is an inconsistency in Bode’s law.

There is mixed reaction among the few I have talked to about whether there could be some truth in Bode’s law. I know of three wonderful writings, by Jack Good (1969), Brad Efron (1971), and Persi Diaconis (1978). They provide different perspectives on whether Bode’s law is a coincidence or not. In 2003, Jim Zidek and I enquired if short sequences of numbers usually follow Bode’s law with a suitable α and β. Some short sequences that didn’t behave erratically did give a terrific fit; two that we tried didn’t. The early primes, because there are too many early twin primes; and the gestation periods of ten small, medium, and huge mammals. When we excluded the whale and the elephant, the fit was wonderful.

But I must caution against mindlessly accepting numerical laws, for in The Letters of Noble Woman (Mrs. La Touche of Harristown), in 1908, I find Mrs. La Touche’s humorous umbrage: “There is no greater mistake than to call arithmetic an exact science. There are Permutations and Aberrations discernible to minds entirely noble like mine […]. For example, if you add a sum from the bottom up, and then again from the top down, the result is always different”!

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